Kylun wrote:For 3E, did they ever stat the Kami, or describe what they look like?

Or is it one of those things where Kami are just "there" and never meant to actually be encountered?

I think it was always intended that there is no such thing as "the kami of $sphere_of_influence". Unlike western concepts of pantheism, the Kara-Tur setting assumed numerous kami, each responsible for a very localised and/or specialised concept. As such, character who rely on the kami for their spells would typically call upon dozens of them to aid them in memorising their spells, often different kami for the same spell as they travel cross-country.

In the setting, it makes no sense to try to "defeat" them, since a mere mortal can't ascend to kami-hood in the same way that a character in certain quasi-European settings might aspire to godhood. And destroying one kami would not meaningfully affect the spells that a caster living there might be able to acquire, since the kami are so common. If you were to try, or even succeed, all you'd really achieve at a campaign level is to make all kami angry at you.

That said, the following Oriental Adventures critters can be used to stat out a kami where needed:

Since it sounds like the presentation of Kami here is pretty close to that used in the Kamigawa setting for Magic the Gathering, I'll bring that up as a counterpoint to the idea that there isn't a reason for fighting against the spirit world. Kamigawa's story revolves around an Emperor who kidnaps a particular spirit, the "child" of the "leader" of Kami-kind, in order to use the trapped spirit's energies to become immortal. Outraged by this violation of their long-standing balance with the material world, the Kami begin to manifest as monsters and attack the mortal races. You could definitely use a plot point like this in a D&D game, and you can justify using any monster that exists with no further explanation than "this is the form which a Kami has chosen to manifest in when doing battle".